Problem 6 - A First Course in Probability

Question

A fair die is rolled 10 times. Calculate the expected sum of the 10 rolls.

Step-by-Step Solution

Answer

The expected sum of 10 rolls of a fair die is 35.

Step 1: Determine the expected value of a single roll

The expected value of a single roll is the average value of all possible outcomes. In the case of a fair six-sided die, the possible outcomes are the numbers 1 through 6, and the probability of each outcome is 1/6. The expected value can be calculated using the following formula: Expected Value = Σ (value × probability) For a fair die, the expected value is: Expected Value = \(1(\frac{1}{6}) + 2(\frac{1}{6}) + 3(\frac{1}{6}) + 4(\frac{1}{6}) + 5(\frac{1}{6}) + 6(\frac{1}{6})\)

Step 2: Calculate the expected value of a single roll

Using the above formula, we can now calculate the expected value of a single roll: Expected Value = \(1(\frac{1}{6}) + 2(\frac{1}{6}) + 3(\frac{1}{6}) + 4(\frac{1}{6}) + 5(\frac{1}{6}) + 6(\frac{1}{6})\) Expected Value = \(\frac{1}{6} + \frac{2}{6} + \frac{3}{6} + \frac{4}{6} + \frac{5}{6} + \frac{6}{6}\) Expected Value = \(\frac{21}{6}\) Expected Value = 3.5

Step 3: Calculate the expected sum of 10 rolls

Now that we know the expected value of a single roll (3.5), we can multiply it by the number of rolls (10) to find the expected sum of the 10 rolls: Expected Sum = (Expected Value of a Single Roll) × (Number of Rolls) Expected Sum = 3.5 × 10 = 35 So, the expected sum of 10 rolls of a fair die is 35.